Plastic Hinge — Definition, Plastic Moment & Collapse Analysis

A plastic hinge is a fully yielded region of a flexural member where the entire cross-section has reached the yield stress Fy, and the section rotates while maintaining a constant resisting moment equal to the plastic moment capacity Mp. Plastic hinges are the foundation of plastic analysis (limit state design), enabling moment redistribution and the formation of collapse mechanisms.

Mp = Fy * Z      Plastic moment capacity
My = Fy * S      Yield moment (first fiber reaches Fy)
Z/S ≥ 1.0        Shape factor — always > 1.0 except for shape at neutral axis

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From First Yield to Full Plastification

As bending moment increases in a simply supported beam of rectangular cross-section:

1. M < My: Entire section elastic. Stress varies linearly from neutral axis.
2. M = My = Fy*S: Extreme fiber reaches Fy. Elastic limit.
3. My < M < Mp: Plastic zone spreads inward from extreme fibers. Elastic core remains at center.
4. M = Mp = Fy*Z: Entire section yielded. Plastic neutral axis (PNA) divides tension and compression zones equally. Moment cannot increase — the hinge rotates.

For a doubly-symmetric W-shape, the transition from My to Mp involves yielding progressing through the flange thickness and into the web. The ratio Z/S (shape factor) is typically:

• W-shapes: 1.10-1.18
• Solid rectangle: 1.50
• Solid circle: 1.70
• Round HSS: ~1.27

Plastic Hinge Rotation Capacity

For plastic analysis to be valid, hinges must possess sufficient rotation capacity before local buckling or fracture. AISC 360 Section B3 requires compact sections (λ ≤ λp for flanges and webs) for plastic analysis. Compact sections can sustain plastic rotations of 3-8 times the yield rotation without strength degradation.

Rotation capacity depends on:

• Flange slenderness: bf/2tf ≤ 0.38*sqrt(E/Fy) for compact (AISC Table B4.1b)
• Web slenderness: h/tw ≤ 3.76*sqrt(E/Fy) for compact
• Lateral bracing: Lb ≤ Lpd to prevent LTB before hinge rotation (AISC 360 Eq. D1-4)
• Axial load: High axial force (P/Py > 0.15) reduces Mp and rotation capacity

Collapse Mechanism Formation

A statically indeterminate structure becomes a mechanism (collapses) when the number of plastic hinges equals the degree of static indeterminacy plus one:

n_h = n_i + 1

Example — Fixed-end beam (indeterminate to 2nd degree):

• 3 hinges needed: left support, right support, midspan
• First hinge forms at the location of maximum elastic moment
• Subsequent hinges form as moment redistributes
• Collapse when the 3rd hinge forms

Example — Two-span continuous beam (indeterminate to 1st degree):

• 2 hinges needed: typically one over the interior support, one in the longer span
• First hinge at the interior support (maximum negative moment)
• Section past this hinge can continue to carry additional load
• Second hinge in the span → mechanism → collapse

Virtual Work Method (Upper Bound Theorem)

The collapse load factor λc is found by equating external work to internal plastic work:

λc * Σ(Wi * δi) = Σ(Mpj * θj)

Where Wi are applied loads moving through displacements δi, and Mpj are plastic moments at hinge locations rotating through angles θj. The lowest λc among all possible mechanisms is the true collapse load (upper bound theorem).

Frequently Asked Questions

What is the difference between a plastic hinge and a real hinge? A real hinge (pin connection) transmits zero moment. A plastic hinge transmits constant moment Mp while rotating — it resists moment up to its plastic capacity. When the applied moment is removed, the plastic hinge ceases to rotate. The analogy is that a plastic hinge behaves like a frictionless hinge with a constant resisting moment.

Why is compactness required for plastic design? Non-compact or slender sections buckle locally before reaching Mp or before developing sufficient plastic rotation. The flange may buckle at a stress below Fy, or the web may buckle under the plastic compression block. Compactness limits (λ ≤ λp) ensure the section can reach Mp and sustain rotations of at least 3θy (yield rotation) without strength degradation.

Can plastic hinges form under axial load? Yes, but the plastic moment capacity reduces. At low axial loads (P/Py < 0.15), the reduction is negligible. At higher axial loads, the plastic neutral axis shifts to accommodate the axial force, reducing the available moment capacity. AISC 360 provides interaction equations (H1-1a/b) for beam-columns. For beam-column plastic analysis, the reduced plastic moment Mp,red = Mp * [1 - (P/Py)²] is a common approximation.

International Code References

• AISC 360: Plastic analysis permitted per Section B3 when sections are compact and laterally braced at hinge locations. Mp per F2.1. Beam-column interaction per H1.
• EN 1993-1-1: Plastic global analysis (Clause 5.4) requires Class 1 cross-sections at plastic hinge locations. Mpl,Rd = Wpl * fy / γM0. Rotation capacity checks per Clause 5.6.
• AS 4100: Plastic analysis per Section 4.5. Compact sections required. Mp per Clause 5.2.
• CSA S16: Plastic analysis per Clause 13.4. Class 1 sections required (Clause 10.1). Mp = φZFy.

Educational reference only. Plastic analysis must be performed under the supervision of a licensed Structural Engineer. Seismic applications (AISC 341) impose additional rotation capacity and connection qualification requirements. All designs must be independently verified.

Disclaimer: This content is for educational purposes only. Results must be verified by a licensed professional engineer. Steel Calculator provides preliminary design tools — NOT a substitute for professional engineering judgment.